Write to a closed field if it is empty

New users specify their name when registering. They can't fix it on their own, the field is closed. Prompt as to old users to give the chance to bring the name. They have an empty name field and no access either.

The name field should be available within their profile edit page. If it's not then ensure the name field is set to display on profile edit and isn't set to read only within CB > Field Management. Note depending on settings your site maybe using First and Last Name fields instead of the single Name field.

You did not understand me. I specifically closed the option to overwrite the name field. The user should not be able to change their data.

I assumed that if you create an additional field and / or script that would check the empty name field or not. If empty, information from the additional field was written to the name field. If the name has already been entered, no record has been made.

Maybe there are some solutions?

There is no option for "if empty, allow edit" specifically. Only way to do that is probably CB Conditional and try having the field condition it self.

I tried the Conditional plugin. Set "Hide field", then specified the field itself, and then the condition "Not empty" and display in "Profile edit". Previously made the field available. As a result, the field, if there is no data, is shown, data entry is impossible - the first letters are deleted when entering text, the field becomes empty.

That's the value resetting functionality. That can be disabled by editing CB Conditional within CB > Plugin Management and adjusting its parameters. Once done see if your usecase then works. Currently it's not possible to toggle off value resetting on a per condition basis so this will impact all conditions.